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Antonio Alarcon

Publications and source records attributed to Antonio Alarcon.

At least 37 records · Page 2Linked to original sources
arXiv2022

Holomorphic Legendrian curves in $\mathbb{CP}^3$ and superminimal surfaces in $\mathbb S^4$

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective $3$-space $\mathbb{CP}^3$, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into $\mathbb{CP}^3$ is path connected. We also show that holomorphic Legendrian immersions from Riemann surfaces of finite genus and at most countably many ends, none of which are point ends, satisfy the Calabi-Yau property. Coupled with the Runge approximation theorem, we infer that every open Riemann surface embeds into $\mathbb{CP}^3$ as a complete holomorphic Legendrian curve. Under the twistor projection $π:\mathbb{CP}^3\to \mathbb S^4$ onto the $4$-sphere, immersed holomorphic Legendrian curves $M\to \mathbb{CP}^3$ are in bijective correspondence with superminimal immersions $M\to\mathbb S^4$ of positive spin according to a result of Bryant. This gives as corollaries the corresponding results on superminimal surfaces in $\mathbb S^4$. In particular, superminimal immersions into $\mathbb S^4$ satisfy the Runge approximation theorem and the Calabi-Yau property.

math.DG↗
arXiv2022

Wild holomorphic foliations of the ball

We prove that the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ admits a nonsingular holomorphic foliation $\mathcal F$ by closed complex hypersurfaces such that both the union of the complete leaves of $\mathcal F$ and the union of the incomplete leaves of $\mathcal F$ are dense subsets of $\mathbb{B}_n$. In particular, every leaf of $\mathcal F$ is both a limit of complete leaves of $\mathcal F$ and a limit of incomplete leaves of $\mathcal F$. This gives the first example of a holomorphic foliation of $\mathbb{B}_n$ by connected closed complex hypersurfaces having a complete leaf that is a limit of incomplete ones. We obtain an analogous result for foliations by complex submanifolds of arbitrary pure codimension $q$ with $1\le q<n$.

math.CV↗
arXiv2020

Algebraic approximation and the Mittag-Leffler theorem for minimal surfaces

In this paper, we prove a uniform approximation theorem with interpolation for complete conformal minimal surfaces with finite total curvature in the Euclidean space $\mathbb{R}^n$ $(n\ge 3)$. As application, we obtain a Mittag-Leffler type theorem for complete conformal minimal immersions $M\to\mathbb{R}^n$ on any open Riemann surface $M$.

math.DG↗
arXiv2020

Complete complex hypersurfaces in the ball come in foliations

In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ is a level set of a noncritical holomorphic function on $\mathbb{B}_n$ all of whose level sets are complete. This shows that $\mathbb{B}_n$ admits a nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces and, what is the main point, that every hypersurface in $\mathbb{B}_n$ of this type can be embedded into such a foliation. We establish a more general result in which neither completeness nor smoothness of the given hypersurface is required. Furthermore, we obtain a similar result for complex submanifolds of arbitrary positive codimension and prove the existence of a nonsingular holomorphic submersion foliation of $\mathbb{B}_n$ by smooth complete closed complex submanifolds of any pure codimension $q\in\{1,\ldots,n-1\}$.

math.CV↗
arXiv2019

The Calabi-Yau problem for Riemann surfaces with finite genus and countably many ends

In this paper, we show that if $R$ is a compact Riemann surface and $M=R\setminus\,\bigcup_i D_i$ is a domain in $R$ whose complement is a union of countably many pairwise disjoint smoothly bounded closed discs $D_i$, then $M$ is the complex structure of a complete bounded minimal surface in $\mathbb R^3$. We prove that there is a complete conformal minimal immersion $X:M\to\mathbb R^3$ extending to a continuous map $X:\overline M\to\mathbb R^3$ such that $X(bM)=\bigcup_i X(bD_i)$ is a union of pairwise disjoint Jordan curves. This extends a recent result for bordered Riemann surfaces.

math.DG↗
arXiv2018

Interpolation and optimal hitting for complete minimal surfaces with finite total curvature

We prove that, given a compact Riemann surface $Σ$ and disjoint finite sets $\varnothing\neq E\subsetΣ$ and $Λ\subsetΣ$, every map $Λ\to \mathbb{R}^3$ extends to a complete conformal minimal immersion $Σ\setminus E\to \mathbb{R}^3$ with finite total curvature. This result opens the door to study optimal hitting problems in the framework of complete minimal surfaces in $\mathbb{R}^3$ with finite total curvature. To this respect we provide, for each integer $r\ge 1$, a set $A\subset\mathbb{R}^3$ consisting of $12r+3$ points in an affine plane such that if $A$ is contained in a complete nonflat orientable immersed minimal surface $X\colon M\to\mathbb{R}^3$, then the absolute value of the total curvature of $X$ is greater than $4πr$.

math.DG↗
arXiv2018

Minimal surfaces in minimally convex domains

In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $M$ into a minimally convex domain $D\subset \mathbb{R}^3$ can be approximated, uniformly on compacts in $\mathring M=M\setminus bM$, by proper complete conformal minimal immersions $\mathring M\to D$. We also obtain a rigidity theorem for complete immersed minimal surfaces of finite total curvature contained in a minimally convex domain in $\mathbb{R}^3$, and we characterize the minimal surface hull of a compact set $K$ in $\mathbb{R}^n$ for any $n\ge 3$ by sequences of conformal minimal disks whose boundaries converge to $K$ in the measure theoretic sense.

math.DG↗
arXiv2018

Interpolation by conformal minimal surfaces and directed holomorphic curves

Let $M$ be an open Riemann surface and $n\ge 3$ be an integer. We prove that on any closed discrete subset of $M$ one can prescribe the values of a conformal minimal immersion $M\to\mathbb{R}^n$. Our result also ensures jet-interpolation of given finite order, and hence, in particular, one may in addition prescribe the values of the generalized Gauss map. Furthermore, the interpolating immersions can be chosen to be complete, proper into $\mathbb{R}^n$ if the prescription of values is proper, and one-to-one if $n\ge 5$ and the prescription of values is one-to-one. We may also prescribe the flux map of the examples. We also show analogous results for a large family of directed holomorphic immersions $M\to\mathbb{C}^n$, including null curves.

math.DG↗
arXiv2018

New complex analytic methods in the theory of minimal surfaces: a survey

In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Riemann-Hilbert boundary value problem. Emphasis is on results pertaining to the global theory of minimal surfaces, in particular, the Calabi-Yau problem, constructions of properly immersed and embedded minimal surfaces in $\mathbb{R}^n$ and in minimally convex domains of $\mathbb{R}^n$, results on the complex Gauss map, isotopies of conformal minimal immersions, and the analysis of the homotopy type of the space of all conformal minimal immersions from a given open Riemann surface.

math.DG↗
arXiv2017

Every meromorphic function is the Gauss map of a conformal minimal surface

Let $M$ be an open Riemann surface. We prove that every meromorphic function on $M$ is the complex Gauss map of a conformal minimal immersion $M\to\mathbb{R}^3$ which may furthermore be chosen as the real part of a holomorphic null curve $M\to\mathbb{C}^3$. Analogous results are proved for conformal minimal immersions $M\to\mathbb{R}^n$ for any $n>3$. We also show that every conformal minimal immersion $M\to\mathbb{R}^n$ is isotopic through conformal minimal immersions $M\to\mathbb{R}^n$ to a flat one, and we identify the path connected components of the space of all conformal minimal immersions $M\to\mathbb{R}^n$ for any $n\ge 3$.

math.DG↗
arXiv2017

Complete densely embedded complex lines in $\mathbb{C}^2$

In this paper we construct a complete injective holomorphic immersion $\mathbb{C}\to\mathbb{C}^2$ whose image is dense in $\mathbb{C}^2$. The analogous result is obtained for any closed complex submanifold $X\subset \mathbb{C}^n$ for $n>1$ in place of $\mathbb{C}\subset\mathbb{C}^2$. We also show that, if $X$ intersects the unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$ and $K$ is a connected compact subset of $X\cap\mathbb{B}^n$, then there is a Runge domain $Ω\subset X$ containing $K$ which admits a complete holomorphic embedding $Ω\to\mathbb{B}^n$ whose image is dense in $\mathbb{B}^n$.

math.CV↗
arXiv2017

Darboux charts around holomorphic Legendrian curves and applications

In this paper, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold $(X,ξ)$. By using such a chart, we show that every holomorphic Legendrian immersion $R\to X$ from an open Riemann surface can be approximated on relatively compact subsets by holomorphic Legendrian embeddings, and every holomorphic Legendrian immersion $M\to X$ from a compact bordered Riemann surface is a uniform limit of topological embeddings $M\hookrightarrow X$ such that $\mathring M\hookrightarrow X$ is a complete holomorphic Legendrian embedding. We also establish a contact neighborhood theorem for isotropic Stein submanifolds, and we find a holomorphic Darboux chart around any contractible isotropic Stein submanifolds in an arbitrary complex contact manifold.

math.CV↗
arXiv2017

Holomorphic Legendrian curves

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on $\mathbb{C}^{2n+1}$ for any $n\in\mathbb{N}$. We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In particular, we show that every open Riemann surface $M$ admits a proper holomorphic Legendrian embedding $M\hookrightarrow\mathbb{C}^{2n+1}$, and we prove that for every compact bordered Riemann surface $M=\mathring M\cup bM$ there exists a topological embedding $M\hookrightarrow \mathbb{C}^{2n+1}$ whose restriction to the interior is a complete holomorphic Legendrian embedding $\mathring M\hookrightarrow \mathbb{C}^{2n+1}$. As a consequence, we infer that every complex contact manifold $W$ carries relatively compact holomorphic Legendrian curves, normalized by any given bordered Riemann surface, which are complete with respect to any Riemannian metric on $W$.

math.CV↗
arXiv2017

New complex analytic methods in the study of non-orientable minimal surfaces in $\mathbb{R}^n$

The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in $\mathbb{R}^n$ for any $n\ge 3$. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immersions of a given bordered non-orientable surface to $\mathbb{R}^n$ is a real analytic Banach manifold, obtain approximation results of Runge-Mergelyan type for conformal minimal immersions from non-orientable surfaces, and show general position theorems for non-orientable conformal minimal surfaces in $\mathbb{R}^n$. We also give the first known example of a properly embedded non-orientable minimal surface in $\mathbb{R}^4$; a Mobius strip. All our new tools mentioned above apply to non-orientable minimal surfaces endowed with a fixed choice of a conformal structure. This enables us to obtain significant new applications to the global theory of non-orientable minimal surfaces. In particular, we construct proper non-orientable conformal minimal surfaces in $\mathbb{R}^n$ with any given conformal structure, complete non-orientable minimal surfaces in $\mathbb{R}^n$ with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits $n$ hyperplanes of $\mathbb{CP}^{n-1}$ in general position, complete non-orientable minimal surfaces bounded by Jordan curves, and complete proper non-orientable minimal surfaces normalized by bordered surfaces in $p$-convex domains of $\mathbb{R}^n$.

math.DG↗
arXiv2017

Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

Let $X$ be a connected open Riemann surface. Let $Y$ be an Oka domain in the smooth locus of an analytic subvariety of $\mathbb C^n$, $n\geq 1$, such that the convex hull of $Y$ is all of $\mathbb C^n$. Let $\mathscr O_*(X, Y)$ be the space of nondegenerate holomorphic maps $X\to Y$. Take a holomorphic $1$-form $θ$ on $X$, not identically zero, and let $π:\mathscr O_*(X,Y) \to H^1(X,\mathbb C^n)$ send a map $g$ to the cohomology class of $gθ$. Our main theorem states that $π$ is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the divisor of a holomorphic form on $X$ can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstneric and Larusson in 2016.

math.CV↗